Oscillation criteria for selfadjoint differential systems
Author:
William T. Reid
Journal:
Trans. Amer. Math. Soc. 101 (1961), 91106
MSC:
Primary 34.30
MathSciNet review:
0133518
Fulltext PDF Free Access
References 
Similar Articles 
Additional Information
 [1]
N.
I. Achieser and I.
M. Glasmann, Theorie der linearen Operatoren im HilbertRaum,
AkademieVerlag, Berlin, 1954 (German). MR 0066560
(16,596f)
 [2]
J. H. Barrett, Twopoint boundary value problems and comparison theorems for fourthorder selfadjoint differential equations and secondorder matrix differential equations, Technical Summary Report #150, April, 1960, Mathematics Research Center, U. S. Army.
 [3]
G.
D. Birkhoff and M.
R. Hestenes, Natural isoperimetric conditions in the calculus of
variations, Duke Math. J. 1 (1935), no. 2,
198–286. MR
1545876, http://dx.doi.org/10.1215/S0012709435001181
 [4]
Gilbert
A. Bliss, Lectures on the Calculus of Variations, University
of Chicago Press, Chicago, Ill., 1946. MR 0017881
(8,212e)
 [5]
Maxime
Bôcher, Applications and generalizations of
the conception of adjoint systems, Trans. Amer.
Math. Soc. 14 (1913), no. 4, 403–420. MR
1500954, http://dx.doi.org/10.1090/S00029947191315009546
 [6]
Magnus
R. Hestenes, Applications of the theory of quadratic forms in
Hilbert space to the calculus of variations, Pacific J. Math.
1 (1951), 525–581. MR 0046590
(13,759a)
 [7]
Henry
Howard, Oscillation criteria for fourthorder
linear differential equations., Trans. Amer.
Math. Soc. 96
(1960), 296–311. MR 0117379
(22 #8159), http://dx.doi.org/10.1090/S0002994719600117379X
 [8]
K. S. Hu, The problem of Bolza and its accessory boundary value problem (Dissertation, University of Chicago, 1932), Contributions to the Calculus of Variations, University of Chicago Press, 19311932, pp. 361443.
 [9]
Walter
Leighton, The detection of the oscillation of solutions of a second
order linear differential equation, Duke Math. J. 17
(1950), 57–61. MR 0032065
(11,248c)
 [10]
Walter
Leighton and Zeev
Nehari, On the oscillation of solutions of
selfadjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. 89 (1958), 325–377. MR 0102639
(21 #1429), http://dx.doi.org/10.1090/S0002994719580102639X
 [11]
Marston
Morse, Sufficient conditions in the problem of Lagrange with fixed
end points, Ann. of Math. (2) 32 (1931), no. 3,
567–577. MR
1503017, http://dx.doi.org/10.2307/1968252
 [12]
Marston
Morse, Sufficient Conditions in the Problem of Lagrange with
Variable End Conditions, Amer. J. Math. 53 (1931),
no. 3, 517–546. MR
1507924, http://dx.doi.org/10.2307/2371163
 [13]
, The calculus of variations in the large, Amer. Math. Soc. Colloquium Publications, vol. 18, 1934.
 [14]
Zeev
Nehari, Oscillation criteria for secondorder
linear differential equations, Trans. Amer.
Math. Soc. 85
(1957), 428–445. MR 0087816
(19,415a), http://dx.doi.org/10.1090/S00029947195700878168
 [15]
William
T. Reid, A Boundary Value Problem Associated with the Calculus of
Variations, Amer. J. Math. 54 (1932), no. 4,
769–790. MR
1506937, http://dx.doi.org/10.2307/2371102
 [16]
William
T. Reid, An IntegroDifferential Boundary Value Problem, Amer.
J. Math. 60 (1938), no. 2, 257–292. MR
1507311, http://dx.doi.org/10.2307/2371292
 [17]
William
T. Reid, A matrix differential equation of Riccati type, Amer.
J. Math. 68 (1946), 237–246. MR 0015610
(7,446a)
 [18]
William
T. Reid, Oscillation criteria for linear differential systems with
complex coefficients, Pacific J. Math. 6 (1956),
733–751. MR 0084655
(18,898a)
 [19]
William
T. Reid, Adjoint linear differential
operators, Trans. Amer. Math. Soc. 85 (1957), 446–461. MR 0088625
(19,550d), http://dx.doi.org/10.1090/S00029947195700886256
 [20]
William
T. Reid, Principal solutions of nonoscillatory selfadjoint linear
differential systems, Pacific J. Math. 8 (1958),
147–169. MR 0098220
(20 #4682)
 [21]
H.
M. Sternberg and R.
L. Sternberg, A twopoint boundary problem for ordinary
selfadjoint differential equations of fourth order, Canadian J. Math.
6 (1954), 416–419. MR 0061738
(15,874c)
 [1]
 N. I. Achieser and I. M. Glasmann, Theorie der linearen Operatoren im HilbertRaum, AkademieVerlag, Berlin, 1954. MR 0066560 (16:596f)
 [2]
 J. H. Barrett, Twopoint boundary value problems and comparison theorems for fourthorder selfadjoint differential equations and secondorder matrix differential equations, Technical Summary Report #150, April, 1960, Mathematics Research Center, U. S. Army.
 [3]
 G. D. Birkhoff and M. R. Hestenes, Natural isoperimetric conditions in the calculus of variations, Duke Math. J. vol. 1 (1935) pp. 198286. MR 1545876
 [4]
 G. A. Bliss, Lectures on the calculus of variations, University of Chicago Press, 1946. MR 0017881 (8:212e)
 [5]
 M. Bôcher, Applications and generalizations of the concept of adjoint systems, Trans. Amer. Math. Soc. vol. 14 (1913) pp. 403420. MR 1500954
 [6]
 M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. vol. 1 (1951) pp. 525581. MR 0046590 (13:759a)
 [7]
 H. C. Howard, Oscillation criteria for fourthorder linear differential equations, Trans. Amer. Math. Soc. vol. 96 (1960) pp. 296311. MR 0117379 (22:8159)
 [8]
 K. S. Hu, The problem of Bolza and its accessory boundary value problem (Dissertation, University of Chicago, 1932), Contributions to the Calculus of Variations, University of Chicago Press, 19311932, pp. 361443.
 [9]
 W. Leighton, The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J. vol. 17 (1950) pp. 5762. MR 0032065 (11:248c)
 [10]
 W. Leighton and Z. Nehari, On the oscillation of solutions of selfadjoint linear differential equations of the fourth order, Trans. Amer. Math. Soc. vol. 89 (1958) pp. 325377. MR 0102639 (21:1429)
 [11]
 M. Morse, Sufficient conditions in the problem of Lagrange with fixed end points, Ann. of Math. vol. 32 (1931) pp. 567577. MR 1503017
 [12]
 , Sufficient conditions in the problem of Lagrange with variable end conditions, Amer. J. Math. vol. 53 (1931) pp. 517546. MR 1507924
 [13]
 , The calculus of variations in the large, Amer. Math. Soc. Colloquium Publications, vol. 18, 1934.
 [14]
 Z. Nehari, Oscillation criteria for secondorder linear differential equations, Trans. Amer. Math. Soc. vol. 85 (1957) pp. 428445. MR 0087816 (19:415a)
 [15]
 W. T. Reid, A boundary value problem associated with the calculus of variations, Amer. J. Math. vol. 54 (1932) pp. 769790. MR 1506937
 [16]
 , An integrodifferential boundary value problem, Amer. J. Math. vol. 60 (1938) pp. 257292. MR 1507311
 [17]
 , A matrix differential equation of Riccati type, Amer. J. Math. vol. 68 (1946) pp. 237246; also Addendum, ibid., vol. 70 (1948) p. 460. MR 0015610 (7:446a)
 [18]
 , Oscillation criteria for linear differential systems with complex coefficients, Pacific J. Math. vol. 6 (1956) pp. 733751. MR 0084655 (18:898a)
 [19]
 , Adjoint linear differential operators, Trans. Amer. Math. Soc. vol. 85 (1957) pp. 446461. MR 0088625 (19:550d)
 [20]
 , Principal solutions of nonoscillatory selfadjoint linear differential equations, Pacific J. Math. vol. 8 (1958) pp. 147169. MR 0098220 (20:4682)
 [21]
 H. M. Sternberg and R. L. Sternberg, A twopoint boundary problem for ordinary selfadjoint differential equations of fourth order, Canad. J. Math. vol. 6 (1954) pp. 416419. MR 0061738 (15:874c)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
34.30
Retrieve articles in all journals
with MSC:
34.30
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719610133518X
PII:
S 00029947(1961)0133518X
Article copyright:
© Copyright 1961 American Mathematical Society
